JR pass

We will be in Japan for 8 days.

Day 1- Osaka
Day 2- 4 -Tokyo
Day 5 -Kyoto
Day6-7 - Osaka

I was wondering if I should buy a JR pass for the above destinations or should I fly from Osaka to Tokyo. By the way, can the JR pass be used for airport shuttle from KIX to Osaka city?

Since you will be making a round-trip from Osaka – Tokyo back to Osaka, a 7-day Japan Rail Pass will pay off (or just about evens out), particularly if you use Haruka train from/to Kansai airport to Osaka.
By shuttle if you mean the limousine bus, I believe the answer is no, it is not covered.

... and of course if you use JR SHinkansen trains for the above round trip.

When will you travel?

I should buy a JR pass for the above destinations

I don't say you should.
A 7-day Japan Rail Pass would not greatly save your cost.

should I fly from Osaka to Tokyo

Well,
flying at JAL or ANA discount prices between ITM and HND
can save your cost by several thousand yen,
compared to taking the typical JR route.

Day 1- Osaka
Day 2- 4 -Tokyo

Won't you make a side trip to Hakone, for example?

Day 5 -Kyoto

Have you ever visited Kyoto City?
One day would be too short for most travelers to explore Kyoto.

If you visit Kyoto outside the busy tourist season
and Takao area is among your destinations,
you may take advantage of your Japan Rail Pass.

[Simulation]

JR regular tickets
(10-day round-trip basic fare tickets
and Ordinary Class Shinkansen and Limited Express tickets)
: You can choose Shinkansen Nozomi trains.
: You pay 29900 yen
for your rides without seat reservation
of Shinkansen Nozomi or Hikari and Limited Express Haruka, or
30820 to 32420 yen
for your rides with your seats reserved
of Shinkansen Hikari and Limited Express Haruka.

Japan Rail Pass
(7-day Ordinary Class pass)
: You are not permitted to take Shinkansen Nozomi trains.
: You pay 29650 yen + handling fee + shipping charge.

Note
: Shinkansen Nozomi trains depart much more frequently than Hikari trains.

I think,you don't need JR pass.It is possible to go From Tokyo to Osaka by 1500 yen by highway bus. Give it a Try.

I would take out Tokyo completely and keep day 2 to 4 all for Kyoto, with side trips to Nara and Kobe / Himeji.

There are obviously different opinions on this (including not even going to Tokyo), but I think that the bottom line is that the two main options (rail pass or domestic air) are both popular and reasonable. Some people take the train but don�ft get a rail pass for this itinerary. (There are a couple of advantages to foregoing the JR pass and just buying individual train tickets, although I always get the pass myself.) A decade or two ago the air travel option was much less popular but in recent years routes have gotten competitive and fares quite cheap. There are some strong proponents of flying.

However, consider that if you fly, you have to get to and from the airport in both cities, go through security and all that, and possibly deal with luggage restrictions (make sure the baggage allowances will meet your needs). (Note, however, that there are new restrictions to be imposed on luggage on the shinkansen, so this is now a consideration for train travel as well.) Chances of a flight disruption (weather, mechanical problems, etc.) are greater for flying than train. None of these things are terribly daunting, but they are enough to deter me. I always take the shinkansen myself. Also, if I have a rail pass I can use it on quite a bit of other train travel, so it minimizes the number of tickets I have to figure out and buy. (Admittedly the JR pass is not too useful in Kyoto, although there are a couple of important lines there that are operated by JR).

There are numerous threads on this subject, so if you want to get more input and opinions you can dig them up. I think it largely depends on your own personal preferences and travel style.

So the choice isn�ft all that simple or clear-cut, although people do tend to have fairly strong personal preferences.

Three nights Tokyo and then three nights Kyoto. Don't waste so much time on Osaka and changing hotels.
Round trip Osaka to Tokyo - minimal savings, extra time required (messing about with voucher exchange), fewer train options.

If you are trying to save every penny/yen, then fly to Tokyo as soon as you arrive at Kansai (there are 10,000 yen fares for international visitors), and anyway, booking in advance the one way trip is 11,000 yen - I wouldn't spend 14,000 yen each way if travelling outside of peak travel periods. Way less than the rail pass.